In OFDM systems, efficient channel estimation schemes are essential for coherent detection of a received signal. After multi-carrier demodulation, the received signal is typically correlated in two dimensions, in time and frequency. The coherence bandwidth is a measure of how rapidly a channel transfer function varies across frequencies. The ability to estimate the coherence bandwidth (and signal-to-noise level) is a requirement for reliably estimating the channel transfer function (CTF), i.e., channel estimation. Also this information could be transmitted back to the base-station or access point radio resource scheduler. In addition, a timing of a first significant channel path (i.e., a propagation time from the transmitter to the receiver) can be estimated. Usually, this timing estimate is then applied in a receiver in a timing correction loop that makes sure that the correct part of a received signal is extracted for further processing.
The starting point for getting a measure of the coherence bandwidth is to evaluate a frequency correlation function based on noisy samples (cƒ)ƒ of the CTF, e.g., based on the following equation:
                                          ρ            ⋒                    ⁡                      (            l            )                          =                                            1                              N                n                                      ⁢                                          ∑                f                            ⁢                                                c                                      f                    +                    l                                                  ⁢                                  c                  f                  *                                                                                        1                              N                d                                      ⁢                                          ∑                f                            ⁢                                                c                  f                                ⁢                                  c                  f                  *                                                                                        (        1        )            where ƒ denotes a sub-carrier index, and Nn and Nd denotes the number of terms in the summation used in the nominator and denominator, respectively. Ideally, the above equation could be solved based on a true (but unknown) value of the channel transfer function h71. However, in practice only noisy estimates, i.e. cƒ=hƒ+εƒ, are available, where εƒ denotes a noise contribution.
Now, assuming that the noise terms are uncorrelated this means that the estimated frequency correlation approaches
                              ρ          ⋒                =                                            (              l              )                        ⁢                                          〈                                                      h                                          f                      +                      l                                                        ⁢                                      h                    f                    *                                                  〉                                                              〈                                                            h                      f                                        ⁢                                          h                      f                      *                                                        〉                                +                                  σ                  ɛ                  2                                                              =                                    R              ⁡                              (                l                )                                                                    R                ⁡                                  (                  0                  )                                            +                              σ                ɛ                2                                                                        (        2        )            where • denotes an ensemble average and σε2 denotes the noise variance. Here, R(•) denotes the frequency covariance. The ensemble average is defined as the mean of a quantity that is a function of the micro-state of a system (the ensemble of possible states), according to the distribution of the system on its micro-states in this ensemble. From equation (2) it can be derived that the presence of noise makes the normalization of the correlation function incorrect. Only in the case where there is no noise, i.e. σε2=0, a correct value can be obtained. Hence, there is an accuracy problem if the coherence bandwidth of the wireless channel and the signal-to-noise (SNR) level of the received signal are estimated based on noisy samples of the channel transfer function (CTF).
A direct way of estimating the timing is to assume a single path channel. In that case the frequency correlation function is ρ(l)=exp(−j2πdl), where d denote the delay of the single path. Hence, the delay can be estimated as
      d    ⋒    =            -              1                  2          ⁢          π          ⁢                                          ⁢          l                      ⁢          ∠ρ      ⁡              (        l        )            and R(l)=hƒ+lhƒ* can be computed, since hƒ+lhƒ*∝ρ(l). Thus, the frequency correlation function has to be determined up to a (real valued) multiplicative constant. However, when such an estimator is applied on multi-path channels it is not the delay to the first significant path, which is estimated but rather the average delay of the channel impulse response.